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C-Integrability Test for Discrete Equations via Multiple Scale Expansions
In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete...
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Інститут математики НАН України
2010
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/146506 |
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irk-123456789-1465062019-02-10T01:24:35Z C-Integrability Test for Discrete Equations via Multiple Scale Expansions Scimiterna, C. Levi, D. In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. 2010 Article C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30 DOI:10.3842/SIGMA.2010.070 http://dspace.nbuv.gov.ua/handle/123456789/146506 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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English |
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In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. |
| format |
Article |
| author |
Scimiterna, C. Levi, D. |
| spellingShingle |
Scimiterna, C. Levi, D. C-Integrability Test for Discrete Equations via Multiple Scale Expansions Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Scimiterna, C. Levi, D. |
| author_sort |
Scimiterna, C. |
| title |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_short |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_fullStr |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full_unstemmed |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_sort |
c-integrability test for discrete equations via multiple scale expansions |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146506 |
| citation_txt |
C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT scimiternac cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions AT levid cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions |
| first_indexed |
2023-05-20T17:24:57Z |
| last_indexed |
2023-05-20T17:24:57Z |
| _version_ |
1796153237697462272 |